A Minkowski Inequality for Hypersurfaces in the Anti‐de Sitter‐Schwarzschild Manifold

Abstract

Abstract. We prove a sharp inequality for hypersurfaces in the ndimensional Anti-deSitter-Schwarzschild manifold for general n ≥ 3. This inequality generalizes the classical Minkowski inequality for surfaces in the three dimensional Euclidean space, and has a natural interpretation in terms of the Penrose inequality for collapsing null shells of dust. The proof relies on a new monotonicity formula for inverse mean curvature flow, and uses a geometric inequality established by the first author in [3].

Inverse mean curvature flows in the hyperbolic 3-space revisited

“…We show that even in the hyperbolic case when the Hawking mass degenerates, this phenomenon persists. A new strategy relying on the conformal structure and the Sobolev type inequality on the sphere at infinity was devised in [1,2] to tackle this difficulty and to obtain Penrose-Gibbon inequalities in the asymptotically flat case. Such a strategy has been further developed by several authors [5,8] to solve related problems.…”

A note on Weingarten hypersurfaces in the warped product manifold

“…Define v = 1 + |∇ϕ| 2 g S n−1 . Then the same calculation as in Proposition 5 in [5] gives that the second fundamental form of Σ has the expression…”

“…If we assume that the closed embedded hypersurface Σ in M satisfies ∂ r , ν > 0, then Σ can be parametrized by a graph on S n−1 (see [5]):…”

Inverse curvature flows in Riemannian warped products

“…1.5]. (vi) In case p = 1 the estimate (1.3) turned out to be strong enough to obtain geometric inequalities, for example in [5,21,50,67]. We are optimistic that Theorem 1.3 will be helpful with such applications as well.…”